At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 18 minutes and a standard deviation of 3 minutes. Using the empirical rule, what percentage of customers have to wait between 12 minutes and 24 minutes?
To calculate the percentage of customers who have to wait between 12 and 24 minutes using the empirical rule, we need to find the z-scores corresponding to 12 and 24 minutes and then calculate the area under the normal curve between these two z-scores.
First, we calculate the z-score for 12 minutes:
z = (X - μ) / σ
z = (12 - 18) / 3
z = -2
Next, we calculate the z-score for 24 minutes:
z = (X - μ) / σ
z = (24 - 18) / 3
z = 2
By using the empirical rule, we know that approximately 68% of the data falls within 1 standard deviation of the mean, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations.
Since the range of 12 to 24 minutes falls within 2 standard deviations of the mean, we can conclude that approximately 95% of customers have to wait between 12 and 24 minutes for their food.